Thisstudywassuggestedbypreviousworkonthesimulationofevolutionequationswithscale-dependentprocesses,e.g.,wave-propagationorheat-transfer,thataremodeledbywaveequationsorheatequations.Here,westudybothparabolicandhyperbolicequations.WefocusonADI(alternatingdirectionimplicit)methodsandLOD(locallyone-dimensional)methods,whicharestandardsplittingmethodsoflowerorder,e.g.second-order.Ouraimistodevelophigher-orderADImethods,whichareperformedbyRichardsonextrapolation,Crank-Nicolsonmethodsandhigher-orderLODmethods,basedonlocallyhigher-ordermethods.WediscussthenewtheoreticalresultsofthestabilityandconsistencyoftheADImethods.Themainideaistoapplyahigher-ordertimediscretizationandcombineitwiththeADImethods.Wealsodiscussthedis-cretizationandsplittingmethodsforfirst-orderandsecond-orderevolutionequations.ThestabilityanalysisisgivenfortheADImethodforfirst-ordertimederivativesandfortheLOD(locallyone-dimensional)methodsforsecond-ordertimederivatives.Thehigher-ordermethodsareunconditionallystable.Somenumericalexperimentsverifyourresults.
